Polyhedrons or Polyhedra

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Polyhedrons or Polyhedra

• A polyhedron is a solid formed by flat surfaces.
• We are going to look at regular convex
  polyhedrons
• regular refers to the fact that every face,
  every edge length, every facial angle, and every
  dihedral angle (angle between two faces) are
  equal to all the others that constitute the
  polyhedron. 
• convex refers to the fact that all of the sides
  of the shapes are flat planes, i.e., they are not
  concave, or dented in. 

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Characteristics of Regular Convex Polyhedra

• Each face is congruent to all others
• Each face is regular
• Each face meets the others in exactly the same
  way  
• So how many regular polyhedra are there?

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The History of the Platonic Solids

• April 11, 2005

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Video

• Pull out your video chart quiz and fill it in as
  the video is played.
• The answers will not be as obvious as our last
  videos, so pay attention!
• You will need this information for your quiz on
  Friday!

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A History of Platonic Solids

• There are five regular polyhedra that were
  discovered by the ancient Greeks.
• The Pythagoreans knew of the tetrahedron, the
  cube, and the dodecahedron the mathematician
  Theaetetus added the octahedron and the
  icosahedron.

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These shapes are called the Platonic solids,
after the ancient Greek philosopher Plato Plato,
who greatly respected Theaetetus' work,
speculated that these five solids were the shapes
of the fundamental components of the physical
universe
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Tetrahedron

The tetrahedron is bounded by four equilateral
triangles. It has the smallest volume for its
surface and represents the property of dryness.
It corresponds to fire.
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Hexahedron

• The hexahedron is bounded by six squares. The
  hexahedron, standing firmly on its base,
  corresponds to the stable earth.

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Octahedron

• The octahedron is bounded by eight equilateral
  triangles. It rotates freely when held by two
  opposite vertices and corresponds to air.

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Dodecahedron

• The dodecahedron is bounded by twelve equilateral
  pentagons. It corresponds to the universe because
  the zodiac has twelve signs corresponding to the
  twelve faces of the dodecahedron.

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Icosahedron

• The icosahedron is bounded by twenty equilateral
  triangles. It has the largest volume for its
  surface area and represents the property of
  wetness. The icosahedron corresponds to water.

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The Archimedean Solids April 7, 2003
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The 13 Archimedean Solids

• All these solids were described by Archimedes,
  although, his original writings on the topic were
  lost and only known of second-hand. Various
  artists gradually rediscovered all but one of
  these polyhedra during the Renaissance, and
  Johannes Kepler finally reconstructed the entire
  set.

• A key characteristic of the Archimedean
  solids is that each face is a regular polygon,
  and around every vertex, the same polygons appear
  in the same sequence, e.g., hexagon-hexagon-triang
  le in the truncated tetrahedron. Two or more
  different polygons appear in each of the
  Archimedean solids, unlike the Platonic solids
  which each contain only a single type of polygon.
  The polyhedron is required to be convex.

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Truncated Tetrahedron Truncated Octahedron
Truncated Cube Cuboctahedron Great
Rhombicuboctahedron Small Rhombicuboctahedron
Snub Cube Truncated Icosahedron Truncated
Dodecahedron Icosidodecahedron Great
Rhombicosidodecahedron Small Rhombicosidodecahedr
on Snub Dodecahedron
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Truncated Polyhedrons

• The term truncated refers to the process of
  cutting off corners. Truncation adds a new face
  for each previously existing vertex, and replaces
  n-gons with 2n-gons, e.g., octagons instead of
  squares. 

cube
truncated cube
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Snub Polyhedrons

• The term snub can refer to a process of replacing
  each edge with a pair of triangles, e.g., as a
  way of deriving what is usually called the snub
  cube from the cube. The 6 square faces of the
  cube remain squares (but rotated slightly), the
  12 edges become 24 triangles, and the 8 vertices
  become an additional 8 triangles.

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April Project

• http//www.scienceu.com/geometry/classroom/buildic
  osa/index.html
• This is the website that contains directions to
  your April Project Building an Icosahedron.
• Your group is going to build one big Platonic
  solid, the icosahedron.
• You will have two class periods to work together.
• This project is due April 30th.